Ratios and Rates

Hong Kong

Stage 1 - Stage 3

Lesson

A ratio compares the relationship between two values. It compares how much there is of one thing compared to another.

If we want to describe the relationship between the number of blue dots and the number of green dots, we could say that there is $1$1 blue dot for every $3$3 green dots. We could also express this as a ratio, which we would write as $1:3$1:3.

A ratio can express a "part to part" relationship like we did in the dot example above. We can also describe a "part to whole relationship". For example, if I wanted to describe the ratio of green dots to all the dots, I would write it as $3:4$3:4 because there are $3$3 green dots and $4$4 dots in total. The order that the words are written in the question correspond to the order of the values in the ratio so don't jumble them around.

We can also express fractions, decimals and percentages as ratios.

During one day, $5$5 flights were delayed and $21$21 flights were on time.

Write a ratio comparing the number of delayed flights to the number of on time flights.

Think: We can compare the two quantities as a ratio $a:b$`a`:`b`. The order is important, so the first number in the ratio will represent the number of delayed flights.

Do: $5:21$5:21

There are $53$53 grams of sugar in $100$100 grams of chocolate spread.

a) Express the amount of sugar to the total amount of spread as a ratio.

Think: We want to express $53$53 parts to $100$100 parts as a ratio.

Do: $53:100$53:100

b) Express the ratio as a fraction.

Think: We want to express $53$53 out of every $100$100 parts as a fraction.

Do: $\frac{53}{100}$53100

We like to express ratios as whole numbers, using the simplest numbers possible. We simplify ratios in a similar way to how we simplify fractions. If we multiply or divide one side by a number, we do the same thing to the other side, this is because we have to keep the equivalence of the statement intact.

**Evaluate: **Express $15:5$15:5 as a ratio in its simplest form.

**Think:** $5$5 is a factor of both $15$15 and $5$5, so we'll divide both sides by $5$5.

**Do:** $\frac{15}{5}:\frac{5}{5}=3:1$155:55=3:1

Ok, let's look at an example of a ratio that is not made up entirely of whole numbers.

**Evaluate:** Simplify the ratio $1\frac{1}{2}:6$112:6

**Think:** the easiest way to make $1\frac{1}{2}$112 a whole number is to multiply it by $2$2 (which would make $3$3). Remember we also have to multiply $6$6 by $2$2, which would be $12$12. So our ratio would be $3:12$3:12. However, $3$3 is a factor of both these numbers, so we can simplify it further.

**Do:**

$1\frac{1}{2}:6$112:6 | $=$= | $3:12$3:12 |

$=$= | $1:4$1:4 |

Write $32$32 minutes to $3$3 hours as a ratio.